Doubly special quantum and statistical mechanics from quantum $\kappa$-Poincar\'e algebra

TL;DR

This paper explores the quantum and statistical mechanics implications of Doubly Special Relativity, introducing a minimum length scale and analyzing how different algebraic parameters affect thermodynamic properties.

Contribution

It constructs a generalized Newton–Wigner operator within the $ ext{kappa}$-Poincaré framework and examines the resulting quantum and statistical mechanical consequences.

Findings

Existence of a minimum length scale in the theory.

Depending on parameters, systems exhibit maximum temperature or become discrete at high temperatures.

Relations between energy/momentum and wavevector suggest modifications to standard quantum mechanics.

Abstract

Recently Amelino--Camelia proposed a ``Doubly Special Relativity'' theory with two observer independent scales (of speed and mass) that could replace the standard Special Relativity at energies close to the Planck scale. Such a theory might be a starting point in construction of quantum theory of space-time. In this paper we investigate the quantum and statistical mechanical consequences of such a proposal. We construct the generalized Newton--Wigner operator and find relations between energy/momentum and frequency/wavevector for position eigenstates of this operator. These relations indicate the existence of a minimum length scale. Next we analyze the statistical mechanics of the corresponding systems. We find that depending on the value of a parameter defining the canonical commutational algebra one has to do either with system with maximal possible temperature or with the one, which in the high temperature limit becomes discrete.